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Centro Euro-Mediterraneo
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www.cmcc.it
February 2009
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Index
1 Introduction............................................................................................ 4
2 Spatial Interpolation .............................................................................. 6
3 Data characteristics............................................................................... 9
3.1 How reliable are predicted data compared with the observed data?..............9
3.2 Do the characteristics of the data influence the result of interpolation? .....11
3.2.1 Summaries of the available data. .................................................................11
3.2.2 Geometrical configuration of the data...........................................................12
3.2.3 The distribution of the nearest-neighbour distance.......................................12
3.2.4 Linearity of the space between grid points of the COSMO LM output ..........12
3.3 Interpolation characteristics.............................................................................13
3.3.1 Interpolation = downscaling?........................................................................13
3.3.2 Which one of the interpolation methods is more efficient in downscaling?...14
4 Spatial interpolation - methods .......................................................... 17
The Data Available are:................................................................................................17
5 Interpolation Methods Developed in ArcGis...................................... 18
5.1 Inverse Distance Weighted (IDW): ...................................................................18
5.2 Radial Basis Functions (RBFs): .......................................................................19
5.3 Polynomial Interpolation (PI) ............................................................................20
5.4 Kriging................................................................................................................21
5.4.1 Linear kriging (LK) ........................................................................................22
5.4.2 Non-linear kriging (NLK)...............................................................................22
6 Parameterization.................................................................................. 24
7 Results and discussions of the interpolation methods developed in
Arcgis ......................................................................................................... 26
7.1 IDW .....................................................................................................................26
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7.2 Radial Basis functions ......................................................................................29
7.3 Kriging................................................................................................................32
7.4 Polynomial interpolation...................................................................................43
8 Discussion ........................................................................................... 48
9 Conclusions......................................................................................... 50
10 Bibliography...................................................................................... 51
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Abstract
This study is a result of the activity carried out at ISC CMCC Division CIRA. The activity
was dedicated on understanding the different modalities used to manage the discrepancy
between the coarse scale at which the COSMO LM1 delivers output and the scale that is
required for most impact studies.
This study reviews 8 methods of interpolation to be used on digital model output data from
the COSMO LM. The model output data are predicted precipitation on a regular grid
shaped surface with 305 point location and a resolution of 2,8 Km. Most of the
geographical spatial analysis require a continuous data set and this study is designed to
create that surface. This study identifies the best spatial interpolation method to use for the
creation of continuous data for predicted precipitation. ArcGIS was employed as the
software for this study. The following interpolated methods were developed in ArcGis:
Inverse Distance Weight, Radial Basis Function (RBF), Kriging (Ordinary, Simple,
Universal and Disjunctive), Local Polynomial interpolation and Global Polynomial
Interpolation. A statistical measurement of the resultant continuous surfaces indicates that
there is little difference between the estimating ability of the 8 interpolation methods with
RBFs performing better overall.
1 Introduction
Precipitation is one of the most frequently used meteorological parameter in impact
studies. The spatial variability of the precipitation depends not only on the nonperiodic or
periodic behaviour of the general atmosphere but also on the sub-grid scale atmospheric
processes such as cloud formation, turbulences, convection, evaporation etc. (Lorenz,
E.N., 1966: ‘’Nonlinearity, Weather Prediction and climate deduction’’ Final Report,
Statistical forecasting project, 22 pg);
1 The Lokal-Modell (LM) is a nonhydrostatic limited-area atmospheric prediction model. It has been designed for both operational numerical weather prediction (NWP) and various scientific applications on the meso- and meso- scale. The LM is based on the primitive thermo-hydrodynamical equations describing compressible flow in a moist atmosphere. The model equations are formulated in rotated geographical coordinates and a generalized terrain following height coordinate. A variety of physical processes are taken into account by parameterization schemes. The basic version of LM has been developed at the Deutscher Wetterdienst (DWD). The subsequent developments related to LM have been organized within COSMO, the Consortium for Small-Scale Modelling. COSMO aims at the improvement, maintenance and operational application of a non-hydrostatic limited-area modelling system based on the LM. At present, different meteorological services partecipate to COSMO. (For other info see www.cosmo-model.org)
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When these physical processes that are governing the atmosphere are integrated into
digital atmosphere models, thank to parameterisation processing (Zorita and von Storch
1999), the output of these models needs only to fulfil the areal character that is needed in
impact studies, which means a higher resolution.
Therefore, when the impact studies are constructed directly from a digital model output (in
our case COSMO LM ), they are unsuitable because the spatial resolution is too coarse
(von Storch et al. 1993, Palutikof & Wigley 1996). So downscaling techniques were
required to generate input data with a finer spatial/temporal resolution. The final result of
these downscaling techniques is expected to have the characteristic of areal distribution
which means not only data with finer spatial resolution but also data that preserve the
characteristics of the conditions that generated them.
The large range of methods encountered in the studied bibliography, has imposed
problems not only for finding the most suitable method but also because of the limited
range of data that is available to use for these methods.
Therefore for this first period I have compressed my activity in finding a way in which the
predicted output of the COSMO LM can be used as base data for the statistical operation
that compose the downscaling techniques.
From the bibliography studied and due to the logistical and data availability, emerged the
idea of concentrating the activity on the statistical downscaling.
Statistical Downscaling (SD): is a method of obtaining high-resolution
climate/meteorological information from relatively coarse-resolution model.
SD methods establish statistical relations among large-scale variables (predictors) and the
variables on a finer-grid scale (predictands).
This paper presents, therefore, an assessment of a regression-based SD method that has
been widely used for constructing climate/meteorological scenarios for daily precipitation
at local sites using digital model grid point information.
If, in most cases, as predictors were considered sea-level pressure (SLP), geopotential
height (Z), temperature or relative (RH) and specific humidity (SH) etc. and as methods,
re-sampling (analogue methods) or weather generators, this time we have decided on a
simpler approach – spatial interpolation.
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This method is based on the idea that terrain variables and geographical location are used
as predictors of the meteorological/climate variables. This is a plausible idea if we consider
that different morphology and geographical location of a region receive in an equal
different way the general atmospheric circulation and these morphological and
geographical characteristics of a region impose local characteristics to the general
atmospheric circulation. So, having in mind that the radiation fluxes stands at the base of
atmospheric circulation, what we are doing is using the very roots of the factors (surely not
all of them) that produces variability of the radiation fluxes (and by so the variability of the
atmospheric circulation at the local scale) – the geography of relief (morphology, location).
In other words, the different orientation of the slopes or the vicinity of an aquatic basin
(sea, ocean), for example, determine a different dynamic of the atmosphere due to their
different degree of isolation (solar radiation), which activates or not elements of the
boundary layer climate (turbulence and air movement, thermal convectivity etc.). As
consequence this elements are responsible of creating diversity at a local level within the
general circulation.
Though a high number of variables (latitude, longitude, elevation, distance from the
nearest coast, slope, aspect, etc.) are used in the bibliographical studies to create the
areal distribution of the precipitation, the position examined in this initial paper would,
mostly, be that of underling methods, characteristics, limitations and accuracy (at least
from statistical point of view) of the predicted areal precipitation interpolated from the grid
shaped output of the Cosmo LM. So, a low number of variables – latitude an longitude –
were used in order to keep the attention on the ‘’behaviour’’ of areal distribution of
precipitation when interpolated from a regular grid Cosmo Lm output with a various
amount of methods.
It remains for the future work to examine if the number of variable, alone, are inducing an
evident positive or negative evolution in the representation of areal precipitation using
similar data and spatial interpolation characteristics and methods.
2 Spatial Interpolation
Interpolation methods allow creating a surface on the base of sample points and predicting
of values in all point of territory. The justification underlying spatial interpolation is the
assumption that points closer together in space are more likely to have similar values than
points more distant. This observation is known as Tobler’s First Law of Geography
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There are many interpolation methods which produce different results using the same
data. The methods can be divided in two main groups: Regular and Irregular.
In case of using the irregular methods the points are connected at triangles. Each
triangle determines a surface which is used for assessment of values of each point. It is
necessary to have many known points for irregular methods implementation. Because of
this, these methods are not suitable for climate/meteo data interpolation. These methods
are often used for relief modelling – creating Digital Elevation Models (DEM) .
The regular methods for interpolation make close initial points and create a net of
identical rectangles with determined by user sides. In this case two main groups of
methods can be used: deterministic and geostatistical.
� DETERMINISTIC interpolation techniques create surfaces from measured
points, based on either the extent of similarity (e.g., Inverse Distance Weighted) or
the degree of smoothing (e.g., Radial Basis Functions). These techniques do not use
a model of random spatial processes.
Deterministic interpolation techniques can be divided into two groups, global and
local. Global techniques calculate predictions using the entire dataset. Local
techniques calculate predictions from the measured points within neighbourhoods,
which are smaller spatial areas within the larger study area.
a. Global interpolation – uses every control point available to derive an equation or
model, so a change in one input value affects the entire map. In other words global
interpolators determine a single function which is mapped across the whole region.
The global interpolation is a 2 step method:
• identification of a statistical relation between the examined parameter and
the potential explicative factors (latitude, altitude, longitude etc.)
• computation of the values of the unknown points using the known values of
the explicative factors.
b. Local interpolation – uses a sample of control points in estimating an
unknown value, so a change in an input value only affects the result within the
neighbour points. In other words the local method is based on the idea that the
values of the nearest points are similar and the variation of the values is increasing
with the increasing of the distance between the points. The advantage of local
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interpolation is that the values of the known points stay unchanged in the process of
interpolation so it can display the spatial anomalies of a phenomenon. The
disadvantage of local interpolation are that:
• Even if displays the spatial anomalies of a phenomenon this does not
explains causal factors.
• It needs a dense network of points with known values.
Some example of deterministic Interpolation methods:
• Polynomials,
• Spatial join (point in polygon),
• Thiessen-Voronoi polygons,
• Triangular Irregular Networks (TIN) and linear interpolation,
• Bi-linear interpolation,
• Spline,
• Inverse Distance Weighting (IDW),
• Radial basis functions.
Whether one of this deterministic methods can be considered local or global
depends on the parameterization of the search neighbourhood. In other words each
of this interpolation methods becomes global if the function is settled to fit the entire
surface or local if the function is settled to fit specified neighbourhoods. For example
Global Polynomials fits a polynomial to the entire surface, Local Polynomial
interpolation fits many polynomials, each within specified overlapping
neighbourhoods.
� GEOSTATISTICAL interpolation techniques (kriging) utilize the statistical
properties of the measured points. Geostatistical techniques quantify the spatial
autocorrelation among measured points and account for the spatial configuration of
the sample points around the prediction location.
Establishing the theoretical presentation, an amount of question has arisen:
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1. Are the predicted data of the COSMO LM accurate enough to be considered
reliable for areal distribution of precipitation?
2. Do the characteristics of the data influence the result of interpolation?
3. Can the interpolation be considered a downscaling method?
4. Which one of the interpolation method is more efficient in downscaling?
The objective of these questions is to underline in a simpler and communicative manner
the characteristics of the data and the interpolation appropriate for the prediction of areal
precipitation.
3 Data characteristics
3.1 How reliable are predicted data compared with the observed data?
The lack of direct measurement of areal precipitation it is the problem that have been
faced when trying to find input precipitation data for all impact studies involving the ground
phase of the water cycle. Reliable direct areal rainfall measurements can be obtained only
at a very limited spatial scale, while the rainfall process is known to exhibit a high degree
of variability both in space and time.
The areal precipitation has been obtained using 3 methods:
- traditional method, which estimates precipitation at ungauged sites through suitable
interpolation method. These are based on the hypothesis that rainfall estimates at
ungauged sites can be obtained as linear or non-linear combinations of the values
measured at a number of instrumented locations.
- indirect estimates of areal rainfall based on the measurement provided since the late
’60s by ground-based meteorological RADARs and remote sensing devices borne on
satellite platforms, such as RADARs and other sensors.
- physically-based numerical models of the atmosphere — though relying on various
theoretical approximations — provide predictions of temporal accumulation values for
areal rainfall over wide spatial scales.
Anyway, the accuracy of each and every one of this techniques is always questioned: the
traditional method is based only on the assumption that rain gauge measurements can
reliably account for the “true point rainfall”, the indirect method encounters problems of
calibration and validation using historical data and the data from the physically-based
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numerical models relays to much on the quality of the model accuracy and also the output
of such models is not known of recreating in a reliable way the actual precipitation
measurements.
So, our attention was drawn towards making a statistical comparison of the predicted and
observed data in order to see how much the predicted data conserves the statistical
characteristics of the observed data (which is assumed that they are “true point rainfall”).
For this, it was used a simple statistical analysis with the purpose of finding whether data
from several groups have a common mean. That is, to determine whether the groups are
actually different in characteristics.
Based on the idea that a valid statistical comparison between analogue variables can not
only be made at the level of the points but also at the areal level, as revealed in the results
of ISC research activity, has been found that 79% of the average values of observed
precipitation it’s included between the area occupied by the maximum and the average are
of the predicted values of precipitation (fig. 1).
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Figure 1. Comparison between observed and predicted precipitation values
(ISC research activity, 2008)
This establishes that development of the predicted values of precipitation follows in a
satisfactory measure the evolution of the observed precipitation values, which means that,
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to a great extent, the predicted data conserves the statistical characteristics of the
observed data.
3.2 Do the characteristics of the data influence the result of interpolation?
Knowledge about the data that are to be interpolated is critical to decide on an appropriate
interpolation method and to understanding the results produced by the interpolation.
Characteristics of the data (spatial representativeness, measurement accuracy, and
existence of spatial relationships) important to consider in interpolation (Shelly Eberly, et
al. 2004) were concentrated in: the analysis of summaries, geometrical configuration,
distance distribution and linearity of the space between grid points of the data.
3.2.1 Summaries of the available data.
It is important to generate some initial summaries of the available data prior to analysis in
order to obtain a better understanding of its spatial characteristics. Reasonable summaries
include, a histogram of the overall values distribution, and summary statistics such as the
data’s mean, standard deviation, and various percentiles (e.g., minimum, median,
maximum, etc.).
Figure 2. Summaries of the COSMO LM output data
Histogram shows that our precipitation values are not perfectly normally distributed. One of
the crosscheck of normal distribution of data is that mean should be closer to the median.
In our case mean is 3,5mm and median is 2,85mm. Also the data shows an asymmetric
distribution with positive skewness and a leptokurtic character which means that a great
number of observations cluster near the average and the rest of observations are skewed
towards the right.
CountMinMaxMeanStd. Dev.
: 305 : 1,07 : 11,8 : 3,5586 : 2,0793
SkewnessKurtosis1-st QuartileMedian3-rd Quartile
: 1,7678 : 5,8295 : 2,2675 : 2,85 : 4,03
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3.2.2 Geometrical configuration of the data
Some of the studies (Morrissey et al. 1995) have assumed that the standard error of the
data disposed in a network, depends not only on the density (which in our case is almost
0.5 grid points/ Km2) of the points with known values but even on the geometrical
configuration. The overall conclusion of this studies concluded that the uniform network
are the best in the terms of accuracy of the representation of the data in space. As the
output of the Cosmo LM is represented on a regular uniform grid we can establish that in
terms of spatial structure of measured field the error that might occur in the process of
interpolation is minimal.
3.2.3 The distribution of the nearest-neighbour distance
The distribution of the nearest neighbour distance it is considered also important in many
studies. It has been examined as mean of distribution (Smith et al. 1986) or as coefficient
of skewness (Matthew Garcia et al 2008) which is said that constitutes a clustering factor
(CF). In the case of the regular grid, the clustering factor remains undefined because of
the singular value of nearest-neighbour distance (2,8Km).
It was considered though that a CF < 0 (a more distributed network) will produce
interpolation errors by reduced resolution of the precipitation field and that CF > 0
(clustering in the network) will produce errors because of reduced areal representation of
the precipitation field.
In the case of CF= 0, which is the case of a regular grid, it is considered that both the
resolution of the precipitation field and the areal representation of the precipitation filed are
characterized by reduced uncertainty and thus by lower errors of the prediction
3.2.4 Linearity of the space between grid points of the COSMO LM output
The data set required for any two-dimensional spatial interpolation exercise consists of
three variables: the parameter of interest (Z), location in the first spatial dimension (x), and
location in the second spatial dimension (y). The variables x and y are in the COSMO LM
data output, longitude and latitude respectively. Such a coordinate system is subject of the
curvature of the earth’s surface. Statistical spatial interpolation techniques assume some
sort of spatial correlation structure defined with respect to the linear distance between two
points in space. (Yan Yu, Deepak Ganesan, Lewis Girod, Deborah Estrin, Ramesh
Govindan, 2003). Therefore, it is not strictly accurate to calculate the distance between two
points in a longitude by latitude coordinate system using a simple linear distance function.
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But in the case of the output data of the COSMO LM, the spatial domain under study is
small enough in geographic extent (2,8Km) in such the error of calculation will be minimal.
As a conclusion we can say that the output data of the COSMO LM presents the most
desirable characteristics in terms of uniform, geometrical and density distribution of the
data for the purposes of prediction of values trough out the studied territory.
3.3 Interpolation characteristics
3.3.1 Interpolation = downscaling?
Based on the available measurement and modelling approaches and the nature of
application, the tendency adopted, in the bibliographical studies, was to replace the
unavailable areal rainfall observations, at the required space-time scales, with suitable
surrogates based on interpolation or downscaling techniques.
Interpolation is the process of predicting the values of a certain variable of interest at
unsampled locations based on measured values at points within the area of interest
(Burrough and McDonnell 1998).
"Downscaling" is based on the view that regional climate is conditioned by climate on
larger, for instance continental or even planetary, scales. Information is cascaded "down"
from larger to smaller scales (Hans von Storch, 2004). In other words, downscaling is any
process where large (coarse) scale output of models is reduced or made finer.
Practically both are establishing the same relation of transfer function between coarse
scale and finer scale, and the spatial and temporal distance is the variables that have to
solve.
“In general, interpolation is applied when ground-based rain gauge and/ or radar networks
are available, while downscaling is applied when using indirect measurements from other
remote sensing devices, or predicted values from atmospheric models, all of which are
usually available at much coarser scales than those required in most hydrological
applications.’’ (L.G. Lanza, J.A. Ramírez and E. Todini - Stochastic rainfall interpolation
and downscaling, Hydrology and Earth System Sciences, 5(2), 139–143 (2001)).
“However, a sharp distinction cannot be made since interpolation and downscaling can
both be incorporated in one single approach, e.g. in order to exploit jointly the information
content of both remotely sensed and rain gauge data (Fiorucci et al., 2001; Todini, 2001)’’
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3.3.2 Which one of the interpolation methods is more efficient in downscaling?
Several characteristics of spatial interpolation have been considered in order to facilitate
the identification of the most efficient method of transferring the character of the coarse
scale COSMO LM to a finer scale trough interpolation. These characteristics include point-
based versus areal-based, global versus local, exact versus approximate, stochastic
versus deterministic, gradual versus abrupt. A compressive description of this
characteristics has been the subject of a subchapter of a study by Shelly Eberly, Jenise
Swall, David Holland, Bill Cox, Ellen Baldridge,2004, study on which we will draw our
conclusion in the following part of the report.
� Point-based versus Areal-based:
Point-based interpolation methods predict values at specific points in space, based on
the values and locations of other individual points in space. Areal interpolation
methods estimate values for entire zones or areas based on data available for a
different set of zones or areas. Even if areal interpolation methods seems to
correspond with our task of representing precipitation on an areal level, a amount of
downfall – like data shaped in a grid network and even limitation of equipment and
analogue data (areal-based has the form of: surrounding area A, B, C, and D,
estimate the values in area E) - are the subject of our reserve toward choosing this
method.
� Global versus Local:
Global interpolators develop and use a single function that estimates values for the
entire sample area whether local interpolators break the full sample area into smaller
pieces that are each evaluated individually by a particular function. Changing one
input data point - in global interpolation - affects the predictions for the entire area,
whether in local interpolator, affects only those areas that consider that point in the
prediction algorithm. In order to choose one of these methods we have to establish if
the function that is mapped has to use the entire area of concern or has to break up
the area into smaller blocks that are evaluated individually. Due to the characteristics
of precipitation that assumes abrupt spatial variance suitable to their complex
morphology our attention was drawn on local interpolation. Global interpolation may
be useful for interpolating surfaces with gradual variation over the area of interest.
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� Exact versus Approximate:
These two methods are considered when we want to produce a surface that avoids or
not sharp peaks and troughs in the estimated surface. Also these features are
important when there is uncertainty about the accuracy of the measured values (i.e.,
measurement error).
These methods vary, based on whether the predicted surface must include the exact
values of the measured data points (exact interpolation) or not (as in the case of
approximate interpolation).
Even on these characteristics the choice of an interpolation method is driven by the
phenomenon that we want to represent. For areal representation of precipitation an
accurate data that reproduces the variation of the precipitation might request that the
predicted surface has to replicate the measured values exactly - so it would be more
appropriate to use an exact method of interpolation.
� Stochastic versus Deterministic:
Whether methods utilize the concept of randomness is another important
characteristic to consider. Stochastic methods incorporate the idea of randomness
into the interpolation process. These methods, which include kriging, allow the
uncertainty of the predicted values to be calculated. Deterministic methods do not
incorporate statistical probability theory into development of the predictions. Instead,
these methods use mathematical formulas or other relationships to interpolate values.
An example of a deterministic method would be one that derives a predicted value by
a simple averaging of nearby measured points. Inverse Distance Weighted (IDW) is a
deterministic method that uses a weighted average of nearby points with distance
being the only factor influencing calculation of the weight. The advantage of stochastic
methods is the ability to provide estimates of uncertainty for the spatial interpolation
model’s output. Kriging is a stochastic method because it assigns weights based not
only on the distance between surrounding points but also on the spatial
autocorrelation among the measured points, which is determined by modelling the
variability between points as a function of separation distance.
� Gradual versus Abrupt:
Another distinguishing characteristic of spatial interpolators is the smoothness of the
predicted surface that is produced. A gradual interpolator produces a surface with
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gradual, relatively smooth, changes. An abrupt interpolator produces a discontinuous
surface with sharp changes. The proximal “nearest-neighbour” method, which sets
unknown points equal to the nearest measured point, is an example of an abrupt
interpolator.
As a conclusion drawn by the theoretical information, but yet to be sustained by the
practical exercise, we can say that – considering the characteristic of our output data
of the COSMO LM and the complex characteristics of precipitation – the most suitable
method that can be used for downscaling has to be a point-based, local, exact (with
reservation due to the accuracy of measurements), deterministic and abrupt method.
These characteristics of the interpolation method, at the end, are pointing on the
spatial variability of the precipitation that the predicted areal distribution has to include.
So, the characteristics of the interpolation has to use a function that evaluates
individually the samples (local method) on detriment of the global which accentuates
the smoothness of the distribution, also for the accuracy of the variability it has to use
the exact and not the approximate method. Thus, the prediction has not to avoid
sharp peaks and troughs in the estimated surface. Also for the accuracy of the spatial
distribution of the precipitation the areal-based method was overlooked because it
estimates values for entire zones or areas which might lead to a loss of detailed
information reason for which the abrupt method has to be employed in order to
realistically catch the gradual changes within the phenomenon. The deterministic
method it might be the better method because in a regular grid the distances between
the sampled points it has no major contribution. So, assigning weights based on the
distance of the surrounding points or modelling the variability between points as a
function of separation distance as the stochastic method does, do not bring
contributions to the final result. In other words in a regular grid shaped data the spatial
variability of a phenomenon can not be contained by the method that attributes
weights to the neighbouring point based on any function of distance separation but by
the method that gives weights taking into consideration mathematical/statistical
relation between the magnitude of the values.
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4 Spatial interpolation - methods
In the task of choosing the right interpolation method I’ve taking into consideration 3
factors:
• the objective of the activity;
• the availability and characteristics of the data (discussed in a previous part of the
report);
• the software available.
The objective would be to find an appropriate method for transferring the characteristics
from the coarse scale data at which the COSMO LM delivers output to finer scale data that
preserves the nonlinear characteristics2 of the conditions that generated them and retains
a high degree of resemblance to the actual or true precipitation field.
The Data Available are:
- hourly predicted precipitation values from the COSMO LM digital model with the
horizontal resolution of 2.8KM (fig.3) ;
- latitude and longitude of the grid points;
- hourly observed data from the meteorological stations: S. Mauro, Ponte Camerelle,
meteorological station A3 highway ;
Both observed and predicted data were selected for a common period of 24h on 1h step
from 4th March 2005.
The activity was developed in ArcGis software, which allows many techniques of
interpolation. ArcGIS is an integrated collection of GIS software products for handling
spatial data, developed by Environmental Systems Research Institute (ESRI). GIS is a
system designed to capture, store, update, manipulate, analyze, and display the
geographic information. For interpolation of meteorological/climatological parameters the
Geostatistical Analyst extension of the version ArcGis 9.2. were used, which provides a set
of tools to create a continuous surface using deterministic and geostatistical methods. Gis
has been vastly used for areal representation of precipitation: Ahers, B., 2006; Hevesi, J.,
2 ”A time-space spectral analysis of the atmosphere certainly reveals pronounced periodically-varying large-scale motions, but the general nonperiodic behaviour and much of the small-scale structure’’ known in digital models as sub-grid scale processes ‘’are direct results of nonlinearity’’. Lorenz, E.N., 1966: ‘’Nonlinearity, Weather Prediction and climate deduction’’ Final Report, Statistical forecasting project, 22 pg;
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Flint, A. & Istok, J. (1992); Fiorucci, P., La Barbera, P., Lanza, L.G. and Minciardi, R.,
2001, D. Kastelec, M. Dolinar, 2000; Tveito,O.E 2002 etc.
5 Interpolation Methods Developed in ArcGis.
A full description of this characteristics has been the subject of the a studies by Shelly
Eberly, Jenise Swall, David Holland, Bill Cox, Ellen Baldridge,2004;
Figure 3. In blue the grid points of the COSMO LM model (resolution 2.8 km), in yellow
meteo station in red the landslide position.
and ArcGis desktop help, on which we will draw our conclusion in the following part of the
report.
5.1 Inverse Distance Weighted (IDW):
IDW is an exact local deterministic interpolation technique. IDW assumes that the value at
an unsampled location is a distance-weighted average of values at sampled points within a
defined neighbourhood surrounding the unsampled point (Burroughs and McDonnell
1998). In this sense, IDW considers that points closer to the prediction location will have
more influence on the predicted value than points located farther away (Johnston et al.
2001). Specifying a higher power places more weight on the nearer points while a lower
power increases the influence of points that are further away. Using a lower power will
result in a smoother interpolated surface being generated (ArcGis Desktop help).
40550
40600
40650
40700
40750
40800
40850
40900
40950
41000
14400 14450 14500 14550 14600 14650 14700 14750 14800 14850 14900longitudine
latitudine 13
1 1 2
12 1 12
8 13
13 7
1 9 87 6 5 4 3
3 1
1 1 1 1
10 9
76 61
46 75 60
45 30
14 13 138
120 105 90
15
15 15 14 9
14 14 14 14 14 14 14 14
247 232
21 200 18
16
16 16
246 240
231 215
199 183
305 292
291 277
276 262
261 255
83 68 53
38 23
11 98
15 15 17 17
12
16
12 11 11
12 6
19
IDW uses:
)(.)zˆ(s1
0 i
N
i
i sz∑=
= λ
zˆ(s0) is the predicted value at the unsampled location s0..
N is the number of measured sample points within the neighbourhood defined for s0.
iλ are the distance-dependent weights associated with each sample point.
z (si) is the observed value at location si. Weights are calculated using:
∑=
−
−
=N
i
p
i
p
ii
d
d
1
0
0λ
11
=∑=
N
i
iλ
Where:
di0 is the distance between the prediction location s0 and the measured location si.
P is the power parameter that defines the rate of reduction of the weights as distance
increases.
IDW is forced to be an exact interpolator to avoid the division by zero that occurs when
di0 = 0 at the sampled points.
IDW is an extremely fast interpolation method, though it is very sensitive to the presence
of outliers and data clustering. In addition, this method does not provide an implicit
evaluation of the quality of the predictions (Burrough and McDonnell 1998, Johnston et al.
2001).
5.2 Radial Basis Functions (RBFs):
RBF are a series of exact deterministic interpolation techniques that include different basis
functions like thin-plate spline, spline with tension, completely regularized spline,
multiquadric function, and inverse multiquadric spline (ArcGis Desktop help). RBFs can be
seen as the process of fitting a flexible membrane to the data points so the total curvature
of the surface is minimized. Being also an exact interpolator, RBFs are different from IDW
20
because they allow the prediction of points above the maximum measured value and
below the minimum measured value. In other words if IDW is based either on the extent of
similarity the RBFs is based or the degree of smoothing.
The predictor defined by a RBF is a linear combination of N basis functions (one for each
data point in the neighbourhood) of the form:
10
1
0 )( ·)z(s ==
+−=∑ niN
i
i ss ωφω
Where:
φ(r) is a radial basis function.
r = ||si – so|| is the distance between the prediction location s0 and the measured
location si.
{ωi: i = 1, 2, …, n + 1} are the weights to be estimated.
The vector of weights w=(ω1, ω2, … ωn) is calculated by solving the following system of
equations:
=
Φ
+ 00'1
1
1
zw
nω
Where:
Φ is a matrix with i, j th elements corresponding to Φ(||si – so||) for each pair of data
points.
1 is a column vector of ones.
Z is a column vector containing the data points.
ωn+1 is a bias parameter.
5.3 Polynomial Interpolation (PI)
PI is an approximate, deterministic interpolation method that fits a mathematical function to
the measured points. Options range from a first-order polynomial (linear) to a second-order
polynomial (quadratic) to higher-order polynomials (ArcGis ranges from the first up to 10th
order polynomial). The predictive surface is typically generated by using a least-squares
regression fit that minimizes squared differences between the surface and measured
21
points. Because it is an approximate interpolator, the surface is not constrained to going
through the measured points as with RBF interpolation. In addition, because the method
generates the best fit (least squares criterion) between the measured points, it is unlikely
that the fitted line will run outside the minimum or maximum measured value, except once
it goes beyond the measured area (i.e., extrapolation).
There are two types of polynomial interpolation — global and local.
• Global polynomial interpolation fits a polynomial model to the entire surface based
on all measured points.
• Local polynomial interpolation fits multiple polynomials using subsets of the
measured points.
Global polynomial interpolation is more appropriate for a surface that varies slowly over
the area of interest, while local polynomial interpolation captures more of the short-range
variation in addition to the long-range trend. Global polynomial interpolation accounts for
bends in the data — one bend with quadratic, two bends with cubic, and so forth. Surfaces
that do not display a series of bends, however, such as one that increases, flattens out,
and increases again, can be better represented using local polynomial interpolation. Both
the global and local methods produce a gradual predicted surface.
5.4 Kriging
Krigging is an optimal interpolation based on regression against observed z values of
surrounding data points, weighted according to spatial covariance values (Geoff Bohling,
2003).
Geostatistical interpolation methods are stochastic methods, with kriging being the most
well-known representative of this category. Kriging methods are gradual, local, and may or
may not be exact (perfectly reproduce the measured data). Also, they are not by definition
set to constrain the predicted values to the range of the measured values. Similar to the
IDW method, kriging calculates weights for measured points in deriving predicted values
for unmeasured locations. With kriging, however, those weights are based not only on
distance between points, but also the variation between measured points as a function of
distance. The kriging process is composed of two parts — analysis of this spatial variation
and calculation of predicted values.
Spatial variation is analyzed using variograms, which plot the variance of paired sample
measurements as a function of distance between samples. An appropriate parametric
22
model is then typically fitted to the empirical variogram and utilized to calculate distance
weights for interpolation. Kriging selects weights so that the estimates are unbiased and
the estimation variance is minimized. This process is similar to regression analysis in that
a continuous curve is fitted to the data points in the variogram.
Kriging creates a continuous surface for the entire study area using weights calculated
based on the variogram model and the values and location of the measured points. The
analyst has the ability to adjust the distance or number of measured points that are
considered in making predictions for each point. A fixed search radius method will consider
all measured points within a specified distance of each point being predicted, while a
variable search radius method will utilize a specified number of measured points within
varying distances for each prediction.
Because kriging employs a statistical model, there are certain assumptions that must be
met. First, it is assumed that the spatial variation is homogenous across the study area
and depends only on the distance between measured sites. There are different kriging
methods and each has other assumptions that must be met.
Kriging methods are often classified as linear and nonlinear (Moyeed and Papritz, 2002;
Papritz and Moyeed, 1999). There are no formal definitions for linear and nonlinear kriging.
5.4.1 Linear kriging (LK)
LK can be defined as kriging methods that derive the estimation using observed values by
assuming a normal distribution of the samples. Linear kriging may include:
• Simple Kriging,
• Ordinary Kriging and
• Universal Kriging.
5.4.2 Non-linear kriging (NLK)
NLK are those methods that derive predictions based on the transformed values of the
observed data. Nonlinear kriging methods consist of:
• Disjunctive Kriging,
• Indicator Kriging,
• multiGaussian kriging,
• lognormal Ordinary Kriging
23
• Model-Based Kriging.
Nonlinear kriging methods have two major advantages over linear kriging:
1) they were developed to model the conditional distribution of the primary variable (i.e.,
to give an estimate of its probability distribution conditional on the available
information);
2) their estimations should theoretically be more precise when a Gaussian random
process is inappropriate to model the observations.
The basic form of the kriging is (Geoff Bohling, 2003):
[ ])()(.m(u) - (u)*Z)(
1
ααα
αλ umuZun
−=∑=
With
u, uα: location vectors for estimation point and one of the neighbouring data points,
indexed by α
n(u): number of data points in local neighbourhood used for estimation of Z*(u)
m(u), m (uα): expected values (means) of Z(u) and (Z uα);
λα(u): kriging weight assigned to datum Z(uα) for estimation location u; same datum will
receive different weight for different estimation location
The goal is to determine weights, αλ , that minimize the variance of the estimator
( ) ( ) ( ){ }uZuZVaruE −= *2σ
Under the unbiasedness constraint ( ){ } 0)(* =− uZuZE .
The random field (RF) Z(u) is decomposed into residual and trend components,
Z(u)=R(u)+m(u) with the residual component treated as an RF with a stationary mean of 0
and a stationary covariance (a function of lag, h, but not of position, u):
( ){ } 0=uRE
( ){ } (h)Ch)}R(u {R(u))(, R=+×=+ EhuRuRCov
24
The residual covariance function is generally derived from the input semivariogram model,
(h).y - Sill y(h)-(0) C(h)C RR ==
Thus the semivariogram we feed to a kriging program should represent the residual
component of the variable.
The three main kriging variants, simple, ordinary and kriging with a trend (universal), differ
in their treatments of the trend component, m(u).
Resuming the mathematical expressions:
• Simple kriging assumes that there is a known constant mean, that there is no
underlying trend, and that all variation is statistical (Wackernagel, 2003).
• Ordinary kriging is similar except it assumes that there is an unknown constant
mean that must be estimated based on the data and the data have no trend (Clark
and Harper, 2001; Goovaerts, 1997).
• Universal kriging differs from the other two methods in that it assumes that there is
a trend in the surface that partly explains the data’s variations. In other words it is
incorporating the local trend within the neighbourhood search window as a smoothly
varying function of the coordinates. Universal Kriging estimates the trend
components within each search neighbourhood window and then performs Simple
Kriging on the corresponding residuals. This should only be utilized when it is
known that there is a trend in the data.
• Disjunctive Kriging is a nonlinear method that is more general than ordinary kriging.
It considers functions of the data rather than using only the data. It assumes that all
data pairs come from a bivariate normal distribution. The theory of disjunctive
kriging and examples of its practical application are described by Armstrong and
Matheron (1986a; 1986b), Rendu (1980) and Oliver et al. (1996).
6 Parameterization
Spatial interpolation methods in ArcGis have various decision parameters to choose from,
no matter if we consider trend based methods or weight based methods. Based on the
knowledge of the nature of data being sampled and processes involved (in our case
rainfall on 305 locations) some parameters can be fixed before the calculations start. The
choices that are made affect the results of the interpolation.
25
Interpolation methods such as IDW and RBF (weight based methods) require fewer
decisions or parameters to manipulate in comparison to kriging methods. Parameters
important for determining the validity of the surface model include (which may vary by
model); surrounding point weight, neighbourhood search, and anisotropy. IDW and RBF
both have similar parameters for determining the small scale variation involved within the
dataset. Kriging interpolation methods utilize functions such as semivariogram and
covariance to assess the weight given to surrounding data points, based on distance and
direction.
Finding the most suitable weight for IDW or RBF is accomplished easily in
GeoStatistical Analyst 9.2 through the “optimize power” feature or consecutively adjusting
it during the procedure until the lowest RMS was obtained.
A curve is fit (quadratic local polynomial equation) to the points and from the curve, the
power that provides the smallest RMS is determined as the optimal power (ESRI, 2004).
When determining the influence of surrounding data points (i.e. neighbourhood) for
weighing interpolation calculations, careful analysis of the involved parameters is
essential.
The neighbourhood search is used to define the neighbourhood shape and the
constraints of the points within the neighbourhood that will be used in the prediction of an
unmeasured location. Neighbourhood search sizes should be large enough to capture the
variability in the data, but small enough to avoid capturing distant points, which create
reduces spatial autocorrelation with the prediction location, hence jeopardizing the
appropriateness of stationarity (Isaaks and Srivastava, 1989).
Anisotropy is a characteristic of a random process that shows higher autocorrelation in
one direction than another.
Although no measures are known that would or could be universally applied to choose the
optimal set of parameters for kriging, cross-validation (a.k.a. "leaving-one-out" method) is
often used to select an interpolator from different number of candidate (Davis, 1987).
Cross validation provides an array of statistical and graphical outputs for comparison of
different parameters before surface model creation, allowing for manipulation of
parameters if needed.
Among the prediction error output statistics for cross validation of deterministic and
stochastic interpolation methods is the mean prediction error (MPE) and root-mean square
26
(RMS). The RMS statistic is a measurement of how close the predicted values are to the
measured values, in which smaller values are preferred. The mean prediction error
statistic (MPE) is a measure of the bias within the model, which will produce values
centered on zero for unbiased models.
Stochastic methods provide additional statistics as an extra measure of uncertainty and
potential error for the prediction model. The kriging standard error, a statistical measure of
uncertainty in the prediction, is calculated by the square root of the kriging variance. RMS
and MPE values can be “standardized” to account for scale dependence, by dividing the
RMS and the MPE each by the standard prediction error to produce RMS standardized
and MPE standardized. The RMS standardized is a measure of variability in addition to the
kriging standard error, in which RMS standardized values will underestimate the variability
when greater than one and overestimate variability where values are less than one (ESRI,
2004).
The use of cross validation prediction error statistics can be a beneficial tool for finding
differences among interpolation methods, however may fall short of clear determination for
finding the “optimal” interpolation method. In such situations, Pearson’s correlation
coefficient and standard deviation values calculations may be beneficial for interpolation
model determination (Isaaks and Srivastava, 1989).
So our decision in choosing the best method of interpolation was highly influenced by the
cross-validation techniques and in addition Pearson’s correlation coefficient and standard
deviation values.
7 Results and discussions of the interpolation methods developed
in Arcgis
7.1 IDW
This spatial interpolation method has various parameters decision. The descriptions below
include the options used in Geostatistical Analyst extend of ArcGis 9,2.
Parameters include: β - the weighting power (exponent); δ - the smoothing parameter; ρ -
the anisotropy ratio; θ - the anisotropy angle.
The method used to consider best IDW interpolation was optimizing parameters via cross-
validation. The parameters were simultaneously adjusted during the procedure until the
lowest RMS was obtained.
27
At the end, the lowest RMS (0,8272) was obtained using a low power (two) in order to give
influence to the points situated farther away too, and the number of points used for each
cell's calculation was limited at 15 to reduce the risk of errors, because points far from the
cell location where the prediction is being made, might have no spatial correlation. Also
the smoothing parameter was not considered, on benefit of standard option, as it
accentuated ‘’ the bulls eye effect’’ (concentric circles around the measured value at the
locations).
The overall predicted contour map is shown in Figure 4 a. This surface shows low degree
of smoothness, a increased spatial variance and even if the power has a low value, the
prediction displays the effect of the concentration around the values of location still causes
‘’bulls eye effect’’.
Figure 4 b. shows the predicted rainfall by the Cosmo LM plotted against interpolated
values for the same locations. The linear correlation coefficient r=0.72 confirms relatively
good overall agreement between independent (Cosmo LM output) and dependent
(interpolated) values. The fact that the larger values of both variables (predicted COSMO
LM and interpolated values) are associated gives a positive related character of correlation
analysis.
The ability to predict extreme values is an important benchmark in evaluating the
performance of an interpolator. Of the extreme values, the lowest values were predicted
Figure 4 a. Interpolated surface using smooth IDW
28
better with dependent and independent low extremes being, up to some extent, similar but
the highest values were inaccurately predicted.
Figure 4 b. Cross-validation of the IDW interpolated prediction
Figure 4 c. shows the distribution of errors (value dependent minus value independent, or
residual) as a function of the magnitude of independent values (Cosmo LM output).The
residuals seems to have a weak tendency of under prediction of the values, with the
increasing of the independent variables. Also the over predicted residuals of the
dependent variable tend to have the outlier characteristics (residuals that fall far from the
regression line) and have a greater variance compared with the under prediction of the
smaller dependent values.
Figure 4 c. Residuals of the IDW interpolated prediction
29
7.2 Radial Basis functions
RBFs can be seen as the process of fitting a flexible membrane to the data points so the
total curvature of the surface is minimized. Being also an exact interpolator, RBFs are
different from IDW because they allow the prediction of points above the maximum
measured value and below the minimum measured value.
There are five different basis functions:
• Thin-plate spline
• Spline with tension
• Completely regularized spline
• Multiquadric function
• Inverse multiquadric function
Each basis function has a different shape and results in a slightly different interpolation
surface. Also the parameters selected for each of the five basis functions could bring an
amount of changes in the final areal distribution of the independent variable. That is way
the selection of the optimal interpolated prediction, for each of the five basis functions, was
done by optimization of the parameters value until the smallest MPE and RMS was
obtained for each of the interpolated predictions. For the optimization and selection of the
parameter’s values was used an identical approach, for the five different function in order
to set a commune base to start from. In terms of the neighbourhood values ArcGis
presents two functions: standard and smooth. The standard search neighbourhood is
defined by the Ellipse parameters: Angle, Major Semiaxis, and Minor Semiaxis. The
Smooth Interpolation option creates an outer ellipse and an inner ellipse at a distance
equal to the Major Semiaxis multiplied by the Smooth Factor. Prediction to each point uses
data inside each corresponding circle/ellipse (ArcGis Desktop help). For the standard
search neighbourhood after the optimisation of the parameters (angle=0;
neighbourhoods=15; major and minor semiaxis = 0,175), a good fitted surface with the
smallest RMS (table 1) was predicted by the RBFs with completed regularized spline
(RMS=0,177), followed by multiquadric (RMS=0,181) and inverse multiquadric RBFs
(RMS=0,182). But when the smoothing effect was added (the same optimized parameters
as the standard search neighbourhood were used and in addition a smooth factor of 0,5)
the only RBFs available in ArcGis 9.2, the inverse multiquadric predicted the best
interpolated surface, having a RMS equal to 0,1501. Changing the smooth factor by
30
increasing or decreasing it with a pass of 0,1 confirmed the fact that still RBFs with a
smooth factor of 0,5 best predicts the interpolated surface.
Table 1 . Comparison of the RBFs (MPE = mean predicted error, RMS = root mean square
of predicted surface):
Radial Basis Functions MPE RMS
Completely regularized Spline 0,00278 0,1778
Spline with tension 0,0000841 0,2993
Multiquadric 0,001143 0,1811
Inverse multiquadric 0,001982 0,1824
Standard
Thin plate spline 0,0006502 0,3018
Smooth Inverse multiquadric 0,0003563 0,1501
Using the cross-validation of the data (fig. 5b and 5c) obtained by inverse multiquadric
RBFs with smoothing factor and the graphical representation (fig. 6.a) an amount of
characteristics could be underlined. The RBFs interpolation produces:
- a smooth surface prediction (fig.6a) due to the fitting of the interpolation curve trough
the measured sample values while minimizing the total curvature of the surface which
can be translated as - prediction above the maximum and below the minimum
measured values;
31
Figure 5 a. Interpolated surface using smooth inverse multiquadric RBFs
Figure 5 b. Cross-validation of the RBFs interpolated prediction
- an almost perfect correlation between the dependent and independent variables
(r=0,99) due to the fact that RBF is an exact method of interpolation. The higher
correlation of the RBFs, compared with another exact method IDW, might be caused
by the higher smoothing factor.
- best prediction made in the area with small values of precipitation, poor prediction in
the area of large and medium precipitation;
32
Figure 5 b. Residuals of the RBFs interpolated prediction
- a more accentuated underestimated prediction compared with the overestimation of
the small precipitation, characteristic observed even in the area of largest
precipitation;
- an overall slightly underestimate prediction with 155 underestimated values and 150
over estimated and maximum of 11,6 mm compared with the 11,8 mm of independent
variable.
7.3 Kriging
If the other interpolation methods (inverse distance squared, splines, radial basis
functions, triangulation, etc.) estimate the value at a given location as a weighted sum of
data values at surrounding locations, Kriging assigns weights according to a (moderately)
data-driven weighting function, rather than an arbitrary function, but it is still just an
interpolation algorithm and will give very similar results to others in many cases (Isaaks
and Srivastava, 1989).
Three kriging methods have been considerate for the analysis of the best geostatistic
interpolation models that include autocorrelation — that is, the statistical relationships
among the measured points.
Even in this case the cross-validation was used in order to select the most suitable of the
methods. Also a statistical description was brought in, with the purpose of drawing a
complete image of the efficiency of the methods (tab. 2).
33
Kriging
MPE RMS ASE MS RMSS Std.dv Mean. Max. Min. Corel. Coef.(r) Ordinary 0,001845 0,9897 1,583 0,000746 0,6261 1.9 3,56 10,2 1,12 0,66
Simple 0,001118 0,4219 1,108 0,000620 0,3839 1.9 3,55 10,2 1,10 0,92
Universal 0,004583 0,4037 0,427 0,007972 0,9516 1,97 3,56 10,6 1,03 0,95
Disjunctive 0,0119 0,4955 1,289 0,008148 0,3863 1,84 3,57 9,9 1,2 0,94
Cosmo LM output 2,05 3,55 11,8 1.07
Table 2. Cross-validation and statistics of the predicted dependent variable
Cross-validation is used to determine "how good" the model is. The goal should be to have
standardized mean prediction errors (MS) near 0, small root-mean-squared prediction
errors (RMS), average standard error (ASE) near root-mean-squared prediction errors
(RMS), and standardized root-mean-squared prediction errors (RMSS) near 1.
The selection of the optimal interpolated prediction, for each of the three kriging method,
was done by comparing the parameters of the cross-validation characteristics in
conformity with the description:
- a optimal prediction has to be unbiased (centered on the true values). If the prediction
errors are unbiased, the mean prediction error (MPE) should be near zero.
- a optimal prediction has to have a valid assessment of uncertainty of the prediction
standard errors (MS).
Each of the kriging methods gives the estimated prediction kriging standard errors.
Besides making predictions, it is estimated the variability of the predictions from the true
values. It is important to get the correct variability. Examples of estimation of variability:
- If the average standard errors (ASE) are close to the root-mean-squared prediction
errors (RMS), the variability in prediction It is correctly assessed.
- If the average standard errors (ASE) are greater than the root-mean-squared
prediction errors (RMS), it is overestimated the variability of the predictions;
- if the average standard errors (ASE) are less than the root-mean-squared prediction
errors (RMS), it is underestimating the variability in the predictions.
Another way to look at this is to divide each prediction error by its estimated prediction
standard error. They should be similar, on average, so the root-mean-squared
standardized errors should be close to 1 if the prediction standard errors are valid:
- If the root-mean-squared standardized errors (RMSS) are greater than 1, it is
underestimated the variability in the predictions;
34
- if the root-mean-squared standardized errors (RMSS) are less than 1, it is
overestimated the variability in the predictions.
In terms of fitted model, kriging was represented by the Spherical model which was
chosen for all kriging methods. With this option, Kriging uses the mathematical function
specified by the method to fit a line or curve to the semi-variance date in the semi-
variogram. Spherical method seems to better fit the spatial variation of the data set
compared with other methods: Circular, Gaussian, Exponential, Tetraspherical etc. The
RMS was used to validate the best fitting model.
� Ordinary Kriging
For ordinary kriging, rather than assuming that the mean is constant over the entire
domain, it is assumed that it is constant in the local neighbourhood of each estimation
point, that is that ( m u ) = m(u) a for each nearby data value, Z (uα) , that we are
using to estimate Z(u).
The ordinary kriging prediction presents the following characteristics :
- a level of the bias higher compared with the simple kriging’s prediction but much
better compared with the rest of the kriging predictions;
- an overestimated level of variability in prediction (ASE > RMS) but closer to a
correct assessing, compared with the simple kriging methods;
- the prediction tends to under predict large values and over predict small values, as
shown in the table 2 (max. 10,2 mm and min. 1, 12 mm);
- a not so good correlation between the dependent and independent variables
(r=0,66)
- considerable extreme residual values settled between 1,2 mm for over prediction
and - 2,3 mm for under prediction;
- best prediction made in the area with small values of precipitation, poor prediction in
the area of large and medium precipitation;
35
Figure 6 a. Interpolated surface using Ordinary Kriging
- a more accentuated underestimated prediction compared with the overestimation of
the large precipitation;
- in the area of small precipitation the variability of the over estimated prediction is
higher compared with the variability of the under estimated prediction;
Figure 6 b. Cross-validation of the Ordinary kriging interpolated prediction
36
Figure 6 c. Residuals of the Ordinary kriging interpolated prediction
� Simple kriging
For simple kriging, it was assumed that the trend component is a constant and known
mean, m(u) = m.
Characteristics (drawn from tabs. 2 and figs 7 a, b ,c) of prediction with simple kriging:
- Simple kriging prediction presents the lowest bias level, the MPE =0, 00118,
and a standard deviation (1,9) close to that of the independent data ;
- the linear correlation coefficient r=0.92 confirms a very good overall agreement
between independent and dependent variables;
Figure 7 a. Interpolated surface using Simple Kriging
37
- poor assessment of the variability in prediction as the distance between the
ASE and RMS is the highest ;
- the prediction tends to under predict large values and over predict small values,
as shown in the table, with considerable extreme residual range (similar with the
ordinary kriging residuals) ,values settled between 1,2 for over prediction and -2,3 for
under prediction;
- simple kriging prediction presents an overall overestimated prediction with 166
overestimated values and 139 underestimated predicted values;
Figure 7 b. Cross-validation of the Simple kriging interpolated prediction
Figure 7 c. Residuals of the Simple Kriging interpolated prediction
38
� Universal kriging the method known as kriging with a trend is much like ordinary
kriging, except that instead of fitting just a local mean in the neighbourhood of the
estimation point, it fits a linear or higher-order trend in the (x,y) coordinates of the data
points.
The cross validation of the Universal kriging prediction has underlined the following
characteristics:
- in terms of bias, accuracy and variability of the prediction the Universal kriging
used
Figure 8 a. Interpolated surface using Universal Kriging
on the grided Cosmo LM data performs better than the ordinary kriging as it uses
more the characteristics of the geometry of data;
- the relation of correlation between the dependent and independent variable is
explained in a proportion of r= 0,95.
- Universal kriging presents an almost exact level of variability in prediction (ASE
> RMS) compared with the other interpolation methods, being slightly overestimated ;
- the prediction tends to under predict large values and also the small one, as
shown in the table 2 (max. 10,6 mm and min. 1, 03 mm);
39
Figure 8 b. Cross-validation of the Universal kriging interpolated prediction
Figure 8 c. Residuals of the Universal Kriging interpolated prediction
- considerable extreme residual values settled between 1,3 mm for over prediction
and - 2,1 mm for under prediction;
- universal kriging predicts better in the area with small values of precipitation and
makes a more accurate (less variable) underestimation compared with the
overestimation;
- in the area of small precipitation the variability of the over estimated prediction is
higher compared with the variability of the under estimated prediction;
40
� Disjunctive Kriging produces a nonlinear unbiased, distribution-dependent estimator
with the characteristics of minimum variance of errors (Burrough and McDonnell,
1998; Yates et al., 1986).
Considering the functions of the data rather than using only the data the Disjunctive
Kriging predicts, based on the characteristics of the Cosmo LM output:
- a accurate prediction with a low bias (MPE = 0,0119) but a not so good accuracy of
the variability wich is highly overestimated (ASE > RMS);
- disjunctive kriging present also a low standard deviation 1,84 compared with the
independent data (2,05) which shows that the prediction is not taken into
consideration the actual data;
Figure 9 a. Interpolated surface using Disjunctive Kriging
- like all the interpolation method presented even DK under estimates the high and
overestimates the low precipitations but in a more accentuated manner, also a
characteristic of distribution-dependent estimator;
- creates considerable amplitude for residual values settled between 1,2 mm for over
prediction and - 2,7 mm for under prediction;
- the prediction of DK shows a very high agreement with the independent data, the
correlation coefficient being situated at 0,94;
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Figure 9 b. Cross-validation of the Disjunctive kriging interpolated prediction
Figure 9 c. Residuals of the Disjunctive Kriging interpolated prediction
- DK predicts better in the area with small values of precipitation where it makes a
more accurate (less variable) underestimation compared with the overestimation;
- In the area with high values of precipitation DK has a more accentuated tendency of
under prediction.
- In the area with mean values of precipitation DK has the tendency of over
predicting;
As a initial conclusion concerning the best prediction among the kriging models used
in this study, we can say that the best prediction is obtained with simple kriging. On
this assumption we have to take into consideration that the main factor that leaded our
assumption towards this conclusion is the spatial characteristics of the data (geometry
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of the distribution of the data, the distribution of the nearest-neighbour distance,).
Many other interpolation studies that used kriging concluded different based on the
different type of data characteristics:
- ‘’universal kriging results were encouraging and comparable with the subjectively
obtained map’’ (D. Kastelec, 2002), the study was concentrated on mean annual
precipitation on 1 KM regular grid data;
- ‘’Indicator kriging gives better estimates than traditional kriging’’ (X. Sun, M.J.
Manton and E.E. Ebert, 2003), the study is combining rain gauge measurements with
satellite infrared data ;
- ‘’ordinary kriging, the best approach to depict the unique variation within the data
set’’ (Julie Earls.2006); study is concentrated on data obtained on radar (NEXRAD);
The features that made the Simple kriging the best method of interpolation (within the
stochastic methods) lays in the characteristics of the data both at the level of
geometry and at the level of measurements and less on modelling the variability of the
data as a function of separation distance.
From the studied bibliography (Oliver Schabenberger and Carol A. Gotway 2004) it
was found that what brings these features in the interpolation computations as a
decisive selection element is the nugget effect. It can be attributed to measurements
errors or spatial source of variation at distance smaller than the sampling interval. In
other words the nugget effect is simply the sum of the measurement error and
microscale variation. Both the measurements error (defined as accuracy of the
variability) and spatial source of variation (thinking at the regular grid) are at level of
the Coasmo LM output minimal and thus it can be said that the nugget effect is also
minimal.
This is mostly, important, if we consider a property of simply kriging which says that
when a no nugget effect is encountered, then ˆU (si) =U(si). More clearly –
without nugget effect, kriging interpolates, with nugget effect kriging it
smooths. Interpolation means following the independent data (observed or
predicted) closely.
Also It was noted , that a good predictor should be variable, in the sense that it follows
the independent data closely. The simple kriging predictor has an this property.
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Consider predicting at locations where data are actually independent, the predictor
psk(Z;s0) becomes psk(Z; [s1,….,sn]’), and in (the simple kriging predictor equation):
). µ(s)- (Z(s) ')()(' )s(Z;p 1000sk ∑ −+=+= σµλλ ssZ when replacing Cov [Z(s0),Z(s)]=σ’ with [Z(s),Z(s)]=Σ and µ(s0) with µ(s) it is obtained
Z(s).
) µ(s)- (Z(s) µ(s)])s.,,[s (Z,p -1n1sk
=
+=… ∑∑
Where:
z(s) is the predicted value at the unsampled location s0..
λ are the distance-dependent weights
µ(s) mean of the random field
In this way, the simple kriging predictor when is interpolating is taking into
consideration the independent data not only the variability of the data as a function of
separation distance . ‘’ It is an ‘exact’ interpolator’’(Oliver Schabenberger and Carol A.
Gotway 2004).
That is way, when cross-validation is performed, the prediction values of the simple
kriging are most likely centered on the true value (low bias), the mean error ,the RMS
and RMSS of the prediction have the smallest values, the correlation between the
dependent and independent variables is most explained (92%), the residuals have a
small fluctuation range but on the other side the variability of the prediction is less
correctly assessed (simple Kriging is the method with the highest overestimated
variability ASE > RMSPE).
7.4 Polynomial interpolation
� Local Polynomial interpolation is a sensitive to the neighbourhood distance that fits
the specified order (zero, first, second, third, and so on) polynomial using all points
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only within the defined neighbourhood. The neighbourhoods overlap and the value
used for each prediction is the value of the fitted polynomial at the center of the
neighbourhood. (ArcGis Desktop help).
Other theoretical characteristics are:
• Creates a surface from many different polynomial formulas.
• Each is optimized for a specified neighborhood.
• The neighborhood shape, maximum and minimum number of points, and a sector
configuration can be specified.
• The sample points in a neighbourhood can be weighted by their distance from the
prediction location.
• Local Polynomial interpolation maps can capture the short-range variation
Characteristics of the Local polynomial prediction underlined by the cross-validation
method (Fig. 9 a, b, c):
- a smooth surface prediction (fig.9 a) due to the fitting of the interpolation curve
trough the measured sample;
- a good coefficient gradient between the dependent and independent variables
(r=0,83);
- a clear tendency of underestimation of the prediction MPE (-0,0056) and a low
degree of bias;
- the prediction tends to under predict large values (max. of prediction 9,6mm) and
over predict small values (min of prediction 1,3 mm);
- presents a high fluctuation of the residuals situated between -3,2 mm for under
prediction and 1,28 mm for over prediction;
- a higher variability of the over predicted residuals along the trend line in the area of
low precipitation.
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Figure 10 a. Interpolated surface using Local Polynomial interpolation
Figure 10 b. Cross-validation of the Local polynomial prediction
Figure 10 c. Residuals of the Local polynomial interpolated prediction
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� Global Polynomial interpolation fits a smooth surface that is defined by a
mathematical function (a polynomial) to the input sample points. The Global
Polynomial surface changes gradually and captures coarse-scale pattern in the data.
(ArcGis Desktop help)
This method generates a smooth surface that does not have to fit to measured points
and does not use a search neighbourhood. A polynomial is used to fit the surface to
the data, so a first order polynomial would have no bends in it, a second order would
have one bend, etc. This method is best used for data that varies slowly over a
landscape or for looking at general trends. It is sensitive to outliers, especially at the
edge of the area of interest.
ArcGis 9,2 offers 10 different order polynomials that can be used from the Global
Polynomial interpolation Set parameters dialog box.
The selected global polynomial of different order brings changes in the final prediction.
That is way the selection of the optimal order of interpolation, was done by
optimization of the parameters value until the smallest RMS was obtained for each of
the polynomial orders (table 3).
Global Polynomial interpolation (order)
1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th
MPE -0,00051 0,001618 0,004909 -0,00016 0,00163 -0,00867 -0,02183 0,01316 0,01244 0,01712 RMS 1,815 1,795 1,778 1,706 1,587 1,435 1,495 1,111 0,915 0,953
Table 3. Mean error and root-mean-error of the 10 different order global polynomial
prediction
Comparing the RMS of the polynomials with different order has been decided that for
a regular grid data, the best prediction is offered by the 9th order-polynomial.
Characteristics of this global polynomial prediction:
- creates smooth surfaces and identifying long-range trends in the dataset, therefore
the prediction fails to represent the short range variability of the phenomenon.
- being sensitive to outliers, the edges of the predicted area are generating errors.
- presents a high fluctuation of the residuals situated between -4,1 mm for under
prediction and 5,4 mm for over prediction;
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Figure 11 a. Interpolated surface using Global 9th order Polynomial interpolation
- presents a low accuracy of the variability of the prediction;
- also presents a good correlation (r=0,86) of the dependent and independent
variables;
- the over and under prediction has a constant variability no matter of the small or
large independent variable (precipitation);
Figure 11 b. Cross-validation of the Global 9th order polynomial prediction
- the prediction tends to under predict not only the large values (max. of prediction
10,5mm) but even the small values (min of prediction -0.4 mm);
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Figure 11 c. Residuals of the Global 9th order Polynomial interpolated prediction
8 Discussion
All interpolation methods create similar evolution and gradual increases and decreasing of
the predicted area along the high and low independent values. The global similarities
among methods were correlated to the similarity of generated statistics for each method.
On a smaller scale, some qualitative variation did exist across all interpolation methods
(table 4).
Interpolation method MPE RMS
Corel. coef.(r)
Std.dv
Mean. Max. Min.
IDW 0,01413 0,8272 0,72 1,5 3,57 8,6 1,4 RBF 0,000356 0,1501 0,99 2,06 3,55 11,6 1,04 Local Polynomial -0,00565 0,595 0,82 1,7 3,55 9,5 1,3 Global Polynomial 0,01244 0,915 0,86 1,98 3,57 10,6 -0.3 Ordinary Kriging 0,001845 0,9897 0,66 1.9 3,56 10,2 1,12 Simple Kriging 0,001118 0,4219 0,92 1.4 3,55 10,2 1,10 Universal Kriging 0,004583 0,4037 0,95 1,97 3,56 10,6 1,03 Disjunctive kriging 0,0119 0,4955 0,94 1,8 3,57 9,9 1,2
Table 4. Statistical properties of the predictions.
The IDW surface model, based on the extent of smoothing, produced “bulls-eye” patterns,
especially along higher values areas. Thus, even though IDW produced respectable
statistics for both cross validation and validation, it was not considered a suitable final
surface model choice.
The RBF methods, based on the degree of smoothing, are also affected by errors of
prediction and under prediction and over prediction. RBF methods typically produce high
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error or uncertainty in areas where values abruptly changes due to the rubber sheeting
applied to the data but overall, RBFs, statistically, produced the highest quality prediction
statistics compare with all other interpolation methods.
Overall the trend based methods (kriging) have had